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Mirrors > Home > MPE Home > Th. List > prod0 | Structured version Visualization version GIF version |
Description: A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.) |
Ref | Expression |
---|---|
prod0 | ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 12630 | . 2 ⊢ 1 ∈ ℤ | |
2 | nnuz 12903 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
3 | id 22 | . . 3 ⊢ (1 ∈ ℤ → 1 ∈ ℤ) | |
4 | ax-1ne0 11214 | . . . 4 ⊢ 1 ≠ 0 | |
5 | 4 | a1i 11 | . . 3 ⊢ (1 ∈ ℤ → 1 ≠ 0) |
6 | 2 | prodfclim1 15883 | . . 3 ⊢ (1 ∈ ℤ → seq1( · , (ℕ × {1})) ⇝ 1) |
7 | 0ss 4398 | . . . 4 ⊢ ∅ ⊆ ℕ | |
8 | 7 | a1i 11 | . . 3 ⊢ (1 ∈ ℤ → ∅ ⊆ ℕ) |
9 | fvconst2g 7214 | . . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = 1) | |
10 | noel 4330 | . . . . 5 ⊢ ¬ 𝑘 ∈ ∅ | |
11 | 10 | iffalsei 4540 | . . . 4 ⊢ if(𝑘 ∈ ∅, 𝐴, 1) = 1 |
12 | 9, 11 | eqtr4di 2783 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {1})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 1)) |
13 | 10 | pm2.21i 119 | . . . 4 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
14 | 13 | adantl 480 | . . 3 ⊢ ((1 ∈ ℤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
15 | 2, 3, 5, 6, 8, 12, 14 | zprodn0 15927 | . 2 ⊢ (1 ∈ ℤ → ∏𝑘 ∈ ∅ 𝐴 = 1) |
16 | 1, 15 | ax-mp 5 | 1 ⊢ ∏𝑘 ∈ ∅ 𝐴 = 1 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 ∅c0 4322 ifcif 4530 {csn 4630 × cxp 5676 ‘cfv 6549 ℂcc 11143 0cc0 11145 1c1 11146 ℕcn 12250 ℤcz 12596 ∏cprod 15893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-prod 15894 |
This theorem is referenced by: prod1 15932 fprodf1o 15934 fprodcllem 15939 fprodmul 15948 fproddiv 15949 fprodfac 15961 fprodconst 15966 fprodn0 15967 fprod2d 15969 fprodmodd 15985 risefac0 16015 coprmprod 16648 coprmproddvds 16650 prmo0 17024 gausslemma2dlem4 27367 breprexp 34416 bcprod 35483 fprodexp 45125 fprodabs2 45126 mccl 45129 fprodcn 45131 fprodcncf 45431 dvmptfprod 45476 dvnprodlem3 45479 |
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